A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

108 Small Circles on the Sphere.
Hence tan2EP or tan2EA = tan DP. tan CP; therefore the
points C, D are harmonic conjugates to A and B; and therefore, &c.
102. If a spherical chord AB of a small circle Xpass through a
fixed point C, the locus of the intersection of tangents AD, BD is
the harmonic polar of C.
F
P B
Fig. 40.
DEM.-Let P be the spherical centre of X. Join PD, PA, PC
by arcs of great circles, and let fall the perpendicular DF
on PC, produced if necessary. Now because in the spherical
quadrilateral DECF the angles E, F are right, we have, equation (134),
tan PC. tan PF = tan PE. tan PD.= tan2P2.
Hence the proposition is proved.
Cor. 1.-If a variable point move along an arc of a great
circle, its harmonic polar passes through a given point.
Cor. 2.-If C be a fixed point in a small circle X; A C, CB
any two arcs of great circles at right angles to each other; the
chord AB passes through a fixed point.